# A Complete Characterization of Projectivity for Statistical Relational Models

arXiv: https://arxiv.org/abs/2004.10984 · HTML: https://mlsanity.com/paper/2004.10984/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2020.

## Built on by (most cited first, 6 of 6)

- [An asymptotic analysis of probabilistic logic programming, with implications for expressing projective families of distributions](https://mlsanity.com/paper/2102.08777/) (2021)
- [Scaling the weight parameters in Markov logic networks and relational logistic regression models](https://mlsanity.com/paper/2103.15140/) (2021)
- [Probabilities of the Third Type: Statistical Relational Learning and Reasoning with Relative Frequencies](https://mlsanity.com/paper/2202.10367/) (2022)
- [Projectivity revisited](https://mlsanity.com/paper/2207.00625/) (2022)
- [The generalised distribution semantics and projective families of distributions](https://mlsanity.com/paper/2211.06751/) (2022)
- [Understanding Domain-Size Generalization in Markov Logic Networks](https://mlsanity.com/paper/2403.15933/) (2024)

## Abstract (arXiv)

A generative probabilistic model for relational data consists of a family of probability distributions for relational structures over domains of different sizes. In most existing statistical relational learning (SRL) frameworks, these models are not projective in the sense that the marginal of the distribution for size-n structures on induced sub-structures of size k<n is equal to the given distribution for size-k structures. Projectivity is very beneficial in that it directly enables lifted inference and statistically consistent learning from sub-sampled relational structures. In earlier work some simple fragments of SRL languages have been identified that represent projective models. However, no complete characterization of, and representation framework for projective models has been given. In this paper we fill this gap: exploiting representation theorems for infinite exchangeable arrays we introduce a class of directed graphical latent variable models that precisely correspond to the class of projective relational models. As a by-product we also obtain a characterization for when a given distribution over size-k structures is the statistical frequency distribution of size-k sub-structures in much larger size-n structures. These results shed new light onto the old open problem of how to apply Halpern et al.'s "random worlds approach" for probabilistic inference to general relational signatures.